A first hires look at tetration \(\lambda = 1\) and \(b = e\)

[sheldonison]

....
Here is, \(\text{tet}_{1,\sqrt{2}}(z)\) over a large domain. This is tetration base \(b =  \log(2)/2\) with period \(2 \pi i\). I kept it to 100 iterations, which gives about 37 digit accuracy.  Again, I've coded overflows to zero.
...the essential singularities are very quiet; they don't really contribute much. I think it's because we are using a bounded base.

Hey James,
A \(2\pi i\) periodic solution for \(b=\sqrt{2}\) that is analytic and converges would seem to be quite the "cool" thing.  Do you think all of the essential singularities are at \(\pm \pi i\)?  Are there any spaces "between" these singularities, or do they get arbitrarily close together?

Assuming this solution is analytic, then there should also be an analytic 1-cyclic Fourier series connecting this solution to the standard solution from the fixed point of 2.  This deserves some investigation, probably not till next week though.

What if you used a Beta solution whose periodic period is greater than the \(\frac{-2\pi i}{\ln(\ln(2))}\approx17.14i\) period?  Would the fixed point of four show up at \(\approx\Im(8.57i)\), like it does in the conventional solution?